Controllability of Partial Differential Equations Governed by Multiplicative Controls

by Alexander Y. Khapalov

Overview

The goal of this monograph is to address the issue of the global controllability of partial differential equations in the context of multiplicative (or bilinear) controls, which enter the model equations as coefficients. The mathematical models we examine include the linear and nonlinear parabolic and hyperbolic PDE's, the Schrödinger equation, and coupled hybrid nonlinear distributed parameter systems modeling the swimming phenomenon. The book offers a new, high-quality and intrinsically nonlinear methodology to approach the aforementioned highly nonlinear controllability problems.

Product Details

ISBN-13:	9783642124129
Publisher:	Springer Berlin Heidelberg
Publication date:	06/01/2010
Series:	Lecture Notes in Mathematics , #1995
Edition description:	2010
Pages:	284
Product dimensions:	6.10(w) x 9.10(h) x 0.70(d)

Table of Contents

Multiplicative Controllability of Parabolic Equations.- Global Nonnegative Controllability of the 1-D Semilinear Parabolic Equation.- Multiplicative Controllability of the Semilinear Parabolic Equation: A Qualitative Approach.- The Case of the Reaction-Diffusion Term Satisfying Newton’s Law.- Classical Controllability for the Semilinear Parabolic Equations with Superlinear Terms.- Multiplicative Controllability of Hyperbolic Equations.- Controllability Properties of a Vibrating String with Variable Axial Load and Damping Gain.- Controllability Properties of a Vibrating String with Variable Axial Load Only.- Reachability of Nonnegative Equilibrium States for the Semilinear Vibrating String.- The 1-D Wave and Rod Equations Governed by Controls That Are Time-Dependent Only.- Controllability for Swimming Phenomenon.- A “Basic” 2-D Swimming Model.- The Well-Posedness of a 2-D Swimming Model.- Geometric Aspects of Controllability for a Swimming Phenomenon.- Local Controllability for a Swimming Model.- Global Controllability for a “Rowing” Swimming Model.- Multiplicative Controllability Properties of the Schrodinger Equation.- Multiplicative Controllability for the Schrödinger Equation.

Editorial Reviews

From the reviews:

“This book offers a detailed study of controllability of infinite-dimensional control systems described by partial differential equations (PDEs), for which the control input appears in a multiplicative way in the differential equations. The book has features that are not often encountered simultaneously in the rest of the literature … . the book is interesting and will have an impact on the topic of the controllability of infinite-dimensional systems. Rigorous proofs are provided for all the results contained in the book.” (Iasson Karafyllis, Mathematical Reviews, Issue 2011 h)

“In this book, the control of evolution processes governed by partial differential equations is studied. … The book is well-written and a welcome addition to the bookshelf for mathematicians with interest in control theory and also researchers in control engineering.” (Martin Gugat, Zentralblatt MATH, Vol. 1210, 2011)

From the Publisher